I. Riemannian geometry of multidimensional Poisson brackets of hydro- dynamic type. In [1] we developed the Hamiltonian formalism of general one-.
ON POISSON BRACKETS OF HYDRODYNAMIC TYPE B. A. DUBROVIN AND S. P. NOVIKOV
I. Riemannian geometry of multidimensional Poisson brackets of hydrodynamic type. In [1] we developed the Hamiltonian formalism of general onedimensional systems of hydrodynamic type. Now suppose that a system of the type of an ideal fluid (possibly with internal degrees of freedom) is described by a set of field variables u = (ui (x)), i = 1, . . . , N , in a space with coordinates x = (xα ), α = 1, . . . , n. The equations of motion of this system have the form of first order equations, linear in the derivatives: (1)
uit = vjiα (u)ujα ,
i = 1, . . . , N,
ujα ≡ ∂uj /∂xα
(here and later we assume summation over repeated indices). The multidimensional analog of the Poisson brackets of hydrodynamic type considered in [1] has the form (2)
{ui (x), uj (y)} = g ijα (u(x))δα (x − y) = ukα bijα k (u(x))δ(x − y).
We define Hamiltonians of hydrodynamic type as functionals of the form Z (3) H = h(u) dn x, that do not depend on the derivatives uα , uαβ , . . . . As in the one-dimensional case, we have that the class of equations (1) and Hamiltonians (3), and the form of the Poisson brackets (2) are invariant under local transformations of the fields u = u(w) (not containing the derivatives). For each α the coefficient g ijα transforms like the set of components of a tensor of the second rank (metric with superscripts i, j); if these tensors do not degenerate, then the coefficients bijα = g isα Γjα k sk transform like Γiα sk the Christoffel symbols of a differential-geometric connection. Under linear α unimodular changes of the spatial variables xα = cα ˜β , cα βx β = const, det(cβ ) = 1, ijα the quantities g ijα and bk for fixed i, j, k transform like vectors: (4)
˜ijβ bijα = cα β bk . k
g ijα = cα ˜ijβ , βg
Here we consider nondegenerate brackets of the form (2); that is, det g ijα 6= 0, α = 1, . . . , n. (Because of the invariance of (4) it is sufficient to assume that the metric g ijα is nondegenerate for one value of α.) From the results of [1] it follows immediately that for a Poisson bracket of the form (2) and for each α the metric ijα g ijα is symmetric, the connection Γiα and has jk is compatible with the metric g zero curvature and torsion. However, for n > 2 it is in general impossible to reduce simultaneously, by a transformation of the form u = u(w), all the metrics g ijα (u) to Date: Received 20/JUNE/84. 1980 Mathematics Subject Classification. Primary 58F05, 76A99. Translated by E. J. F. PRIMROSE. 1
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constants, and the coefficients of connection to zero. An obstruction is the tensor (in the field variables) (5)
iαβ iα Tjk = Γiβ jk − Γjk .
It is more convenient to consider a tensor with superscripts (6)
jαβ lkb T ijkαβ = −g ipα g kqβ Tpq = bijα − bkjβ g liα , l g l
defined also in the case of degenerate metrics. Example. For Poisson brackets that correspond to the Lie algebra of vector fields in Rn (see [2]) (7)
{pi (x), pj (y)} = pi (x)δj (x − y) − pj (y)δi (y − x),
N = n,
the metrics g ijα and the coefficients bijα have the form k (8)
g ijα = pi δ jα + pj δ iα ,
bijα = δki δ jα . k
Here the simultaneous reduction of metrics to constant form is impossible for n > 2, since T ijkαβ 6= 0. Metrics g ijα of the form (8) are degenerate for N > 2. The same is true for all the examples, considered in [2], of multi-dimensional brackets of hydrodynamic type with more than two fields. Theorem 1. Nondegenerate Poisson brackets of the form (2) with N > 3 fields ijα k can be reduced by a transformation u = u(w) to linear form, g ijα = cijα k u + g0 , ijα ijα ijα α = 1, . . . , n, n > 2, where ck , g0 and bk are constants. We examine separately the cases N = 1 and N = 2. For N = 1 a Poisson bracket of the form (2) is specified by the set g α (u), α = 1, . . . , n, which can be arbitrary functions of the field variable u. Now suppose that N = 2. The conditions on the metric g ijα and the symmetric connections compatible with them for zeros of the curvature bijα = g isα Γiα sk , which follow from the Jacobi identity for the bracket (2), k have the form (9)
T ijkαβ = −T ijkβα = T kijαβ ,
where the tensor T ijkαβ is defined by (6), and also (10)
jlαβ jkαβ Tjikαβ Tm = Tjilαβ Tm ,
where (11)
Tjikαβ = g ilα Tljkαβ .
Let us consider a classification of admissible (in the sense of (9) and (10)) nondegenerate metrics g ijα in the first nontrivial case n = N = 2. Let λ1 and λ2 be the of the pair of quadratic forms g ij1 , g ij2 ; that is, the roots of the equation (12)
det |g ij1 − λg ij2 | = 0.
Theorem 2. Admissible pairs of metrics g ij1 and g ij2 reduce to one of the following canonical forms: a) For nonconstant and noncoincident λ1 , λ2 , the metrics can be written in the coordinates u1 = λ1 , u2 = λ2 in the form (13)
g ij1 = h−2 i δij ,
g ij2 = (ui )−1 h−2 i δij ,
ON POISSON BRACKETS OF HYDRODYNAMIC TYPE
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where the coefficients h1 , h2 satisfy the system of equations ∂ ψ ∂h2 ∂ ψ ∂h1 = −h2 1 √ , = h1 2 √ , ∂u2 ∂u ∂u1 ∂u u1 u2 u1 u2 (14) ∂2ψ ∂ψ ∂ψ 2(u1 − u2 ) 1 2 + − = 0. ∂u ∂u ∂u1 ∂u2 b) λ1 6= const, λ2 = c; in certain coordinates the metrics reduce to the form (15)
g ij1 = h−2 i δij ,
−2 g ij2 = λ−1 i hi δij ,
λ1 = u1 , λ2 = c,
where (16)
" # ∂h1 c(au2 + b) 1 =− p + ψ(u ) h2 , ∂u2 2 u1 (c − u1 )3
r ∂h2 u1 =a h1 , 1 ∂u c − u1
a, b are any constants, and ψ(u1 ) is an arbitrary function. c) λ1 6= λ2 are constant numbers; in certain coordinates the metrics reduce to the form (17)
g ij1 = h−2 i δij ,
−2 g ij2 = λ−1 i hi δij ,
h2 = ∂f (u1 , u2 )/∂u2 ,
h1 = f (u1 , u2 ),
∂ 2 f /∂u1 ∂u2 = f.
d) Coincident eigenvalues λ1 ≡ λ2 , and the metrics are indefinite; in certain coordinates they reduce to the form 0 1 0 µ(u1 ) ij1 ij2 (18) (g ) = , (g ) = , 1 0 µ(u1 ) u2 ϕ(u1 ) + ψ(u1 ) where µ, ϕ and ψ are arbitrary functions. It is not difficult to determinee the degree of arbitrariness in the choice of “canonical” coordinates that reduce admissible pairs of metrics to the form (13), (15), (17) or (18). Metrics in general position (type a)) depend on four “essential” arbitrary functions of one variable (plus an arbitrary change of coordinates), those of type b) and d) on three functions of one variable, and those of type c) on two. The brackets (7) reduce to the form (18). II. Poisson brackets for nonhomogeneous one-dimensional systems of hydrodynamic type. Along with homogeneous systems of hydrodynamic type it is natural to consider the class of nonhomogeneous systems (19)
uit = vji (u)uix + f i (u),
i = 1, . . . , N
(here we restrict ourselves to the one-dimensional case). Correspondingly we introduce the class of nonhomogeneous Poisson brackets of the form (20)
k ij {ui (x), uj (y)} = g ij (u(x))δ 0 (x − y) + [bij k (u)ux + h (u)]δ(x − y),
with respect to which the systems (19) can be written in Hamiltonian form with Hamiltonians of the form (3). Theorem 3. 1) For nonhomogeneous brackets of the form (20) the tensor hij (u) determines the usual (finite-dimensional ) Poisson bracket on functions of u. 2) If the metric g ij (u) is not degenerate, then by a transformation of coordinates u = u(w) the bracket (20) reduces to the form (21)
k ij {wi (x), wj (y)} = g˜ij δ 0 (x − y) + [cij k w + d ]δ(x − y),
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ij where the coefficients g˜ij , cij are constants, the cij k and d k are the structural constants of a semisimple Lie algebra with Killing metric g˜ij , and dij = −dji is an arbitrary cocycle on this Lie algebra.
Example. The equations of the problem of n-waves [3] (22)
Mt − ϕ(Mx ) = [M, ϕ(M )],
where M = (Mij ) is an n × n matrix with zero trace (possibly with additional symmetries), and ϕ(M ) = (λij Mij ), are Hamiltonian with respect to the brackets R (21), dij = 0, with quadratic Hamiltonian 2H = Tr(M ∗ · ϕ(M )) dx. As we know (see [3]), these equations are integrable by the method of the inverse problems for λij = (ai − aj )/(bi − bj ). III. Differential-geometric brackets of higher orders. Definitions. 1. A general homogeneous differential-geometric Poisson bracket of order m on a space of fields uj (x) is a bracket of the form k (m−1) (23) {ui (x), uj (x)} = g ij (u(x))δ (m) (x − y) + bij (x − y) k (u(x))ux δ ij k k l (m−2) + [cij (x − y) + · · · , k (u(x))uxx + dkl (u(x))ux ux ]δ
so that any Hamiltonian of hydrodynamic type (3) generates equations of the form (24)
uit = vji (u)∂ m uj /∂xm + · · · ,
where the right-hand side is a graded-homogeneous polynomial of degree m in the derivatives, where the derivative ∂ k u/∂xk has index k. 2. A nonhomogeneous bracket is the sum of homogeneous Poisson brackets of different orders. According to a conjecture of the authors, the properties of these brackets are as follows: a) if the “metric” g ij is nondegenerate, then the homogeneous bracket of a local variable u(w) reduces to the form g0ij δ (m) (x − y), g0ij = const; and b) for nonhomogeneous brackets with nondegenerate highest term of order m all the homogeneous components of order m − k depend on no more than the kth powers of the original fields uj and their derivatives (in those coordinates uj where the highest term of order m reduces to a canonical constant of the form g0ij δ (m) ). In particular, the term corresponding to k = 1 generates a Lie algebra (cf. Theorem 3 above). Remark. Since the group of translations uit = uix can be a Hamiltonian system only if there is no homogeneous term of the first order in the bracket, it is of particular interest to classify nonhomogeneous brackets with this property. Just as in the case of one special variable n = 1 (see above), this class of brackets admits a natural generalization to the case n > 1. References [1] B. A. Dubrovin and S. P. Novikov, Dokl. Akad. Nauk SSSR 270 (1983), 781–785; English transl. in Soviet Math. Dokl. 27 (1983). [2] S. P. Novikov, Uspekhi Mat. Nauk 37 (1982), no. 5(227), 3–49; English transl. in Russian Math. Surveys 37 (1982). [3] S. P. Novikov (editor), Theory of solitons, “Nauka”, Moscow, 1980; English transl., Plenum Press, New York, 1984. Moscow State University
